EXERCISES in .NET framework

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EXERCISES
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of $5 for each cubic metre of chemical waste that is removed Compute an average cost optimal policy by policy iteration or linear programming 65 A stamping machine produces six-cornered plates of the illustrated form
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a c b a b c
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The machine has three pairs of adjustable knives In the diagram these pairs are denoted by a, b and c Each pair of knives can fall from the correct position during the stamping of a plate The following ve situations can occur: (1) all three pairs have the correct position, (2) only pairs b and c have the correct position, (3) only pair b has the correct position, (4) only pair c has the correct position and (5) no pair has the correct position The probabilities qij that during a stamping a change from situation i to situation j occurs are given by 3 1 4 4 0 0 0 1 1 1 0 2 4 4 0 3 0 1 (qij ) = 0 0 4 4 1 1 0 0 0 2 2 0 0 0 0 1 After each stamping it is possible to adjust the machine such that all pairs of knives have the correct position again The following costs are involved The cost of bringing all pairs of knives into the correct position is 10 Each plate produced when j pairs of knives have the wrong position involves an adjustment cost of 4j Compute a maintenance rule that minimizes the average cost per stamping by policy iteration or linear programming 66 An electricity plant has two generators j = 1 and 2 for generating electricity The required amount of electricity uctuates during the day The 24 hours in a day are divided into six consecutive periods of 4 hours each The amount of electricity required in period k is dk kWh for k = 1, , 6 Also the generator j has a capacity of generating cj kWh of electricity per period of 4 hours for j = 1, 2 An excess of electricity produced during one period cannot be used for the next period At the beginning of each period k it has to be decided which generators to use for that period The following costs are involved An operating cost of rj is incurred for each period in which generator j is used Also, a set-up cost of Sj is incurred each time generator j is turned on after having been idle for some time Develop a policy-iteration algorithm that exploits the fact that the state transitions are deterministic Solve for the numerical data d1 = 20, d2 = 40, d3 = 60, d4 = 90, d5 = 70, d6 = 30, c1 = 40, c2 = 60, r1 = 1000, r2 = 1100, S1 = 500 and S2 = 300 67 Every week a repairman travels to customers in ve towns on the successive working days of the week The repairman visits Amsterdam (town 1) on Monday, Rotterdam (town 2) on Tuesday, Brussels (town 3) on Wednesday, Aachen (town 4) on Thursday and Arnhem (town 5) on Friday In the various towns it may be necessary to replace a certain crucial element in a piece of electronic equipment rented by customers The probability distribution of the number of replacements required at a visit to town j is given by {pj (k), k 0} for j = 1, , 5 The numbers of required replacements on the successive days are independent of each other The repairman is able to carry M spare parts If the number of spare parts the repairman carries is not enough to satisfy the demand in a town, another repairman has to be sent the next day to that town to complete the remaining replacements The cost of such a special mission to town j is Kj At the end of each day the repairman may decide to send for a replenishment of the spare parts to the town where the repairman is The cost of sending such a replenishment to town j is aj Develop a value-iteration algorithm for the
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